Calculating Dependant Probability of Multiple Events

The Problem

-There are 5850 total tickets.
- 8 tickets will be randomly chosen at the same time.
- Once a ticket is chosen, the same ticket number cannot get chosen again.
- I have 290 tickets or 3.57% of the total supply.
- There is a reward structure if you get 3 tickets, 4 tickets, 6 tickets, and 8 tickets you win a prize.
- What is the probability that my 290 tickets get picked 1 time, 2 times, 3 times, 4 times, 5 times, 6 times, 7 times, and all 8 times, and what is the formula to calculate that?

What I have so far

I believe I was able to calculate the probability of getting 1 ticket (290*8/5850 = 28.58%)

I get confused once I need to calculate it happening 2 times and on. Would it be (289*8)/(5850*2) or (289*7)/(5850*2)?
  • Do you mean the following? "There are 5850 tickets, among them 8 are "winning". I have 290 tickets. What is the probability of me having 1, 2, ..., 8 winning tickets?"

  • Yes - if your ticket is chosen then it can be redeemed for a prize thus it would be considered a winning ticket.

  • I accepted but I am unable to understand the formula and I am not gifted in math. What does "!" mean in the equation How did you get 5842? That is 18 minus 5860 which I am not sure where that comes from. Can you help me in more layman terms on how to solve?

  • The ! (factorial) symbol means you have to multiply by all the numbers smaller than that, e.g., 5! = 5*4*3*2*1 = 120. For the other question, 5842 = 5850-8, and you have 5850 tickets total, so that is the number of non-winning tickets.

  • The binomial symbol gives you the way of choosing a subset of a set of a given size. I would recommend giving a look to the Wikipedia page https://en.wikipedia.org/wiki/Binomial_coefficient to get an idea of what's going on.

Answer

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